John Tavares put the rumors to rest Thursday, signing a six-year contract extension to stay with the Islanders for a reported $33 million.

Tavares was the No. 1 overall pick in 2009, and as he approached the end of his entry-level contract it was widely speculated (mostly by his hometown press around Toronto) that the center would leave Long Island.

Instead, Tavares, who turns 21 Tuesday, the fifth day of training camp, made good on his word that he wants to remain with the Islanders.

“I’m thrilled to show the organization and loyal Islanders fans that I want nothing more than to win on Long Island,” Tavares said. “This is where I wanted to live, play and ultimately lead our team to a Stanley Cup.”

Tavares now enters the next half-decade with fellow “young core” members Kyle Okposo and Michael Grabner, who each signed a five-year deal earlier this summer.

“We’ve been known as a hard-working team that competes,” Tavares said. “Now a lot of our talent is starting to come forward.”

Tavares will play this year on his entry-level deal, making $900,000 ($3.75 million salary-cap hit), then will make $4 million next year, $5 million in 2013-14, and $6 million the remaining four years, The Post confirmed.

General manager Garth Snow made it clear this offseason that locking up his young players was a priority, and none was more important than Tavares.

“He is a key component to the core of players we’ve decided to build this team around,” Snow said. “We couldn’t be happier to ensure that he will be a New York Islander for years to come.”

Tavares has always had expectations to live up to, first as the wunderkind from Mississauga, Ontario; then as the first overall pick; now as the man carrying an annual $5.5 million salary-cap hit.

“Charles [Wang] is a great owner and he’s been great to us,” Tavares said. “Once we start winning and things are more successful on the ice, people will to notice us a little more and what we’re all about and where we’re headed.”

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Snow said he would spend the rest of Thursday trying to get Josh Bailey’s contract situation settled. The forward is a restricted free agent, and the two sides have been working all summer in an attempt to reach an agreement.

Because training camp physicals start Friday, Bailey has less than a day to sign or he will start to miss team activities.

“What player wants it to come down to the final day?” Bailey asked after his informal skate at the team complex in Syosset. “There’s not much time left, but I still have faith that we’ll get things resolved.”